every orthonormal set is linearly independent

Proof. We denote by ⟨⋅,⋅⟩ the inner product of L. Let S be an orthonormal set of vectors.
Let us first consider the case when S is finite, i.e.,
S={e1,…,en} for some n.
Suppose

λ1⁢e1+⋯+λn⁢en=0

for some scalars λi (belonging to the field on the
underlying vector space of L). For a fixed k in 1,…,n,
we then have

0=⟨ek,0⟩=⟨ek,λ1⁢e1+⋯+λn⁢en⟩=λ1⁢⟨ek,e1⟩+⋯+λn⁢⟨ek,en⟩=λk,

so λk=0, and S is linearly independent.
Next, suppose S is infinite (countable or uncountable). To prove
that S is linearly independent, we need to show that
all finite subsets of S are linearly independent. Since any
subset of an orthonormal set is also orthonormal, the infinite case
follows from the finite case. □